Given a balance scale that is sure to break after X weighings, find an equation for the largest number of coins N, from which you can determine a fake coin that has the wrong weight if
A: You know whether the fake is lighter or heavier
B: You do not know whether the fake is lighter or heavier
(Assume only one of the N coins is fake)
For part A, everyone seems agreed that n=3^x is the maximum. For part B, you can definitely manage n=3^(x-1). That is, add one more weighing. I don't think this is the best possible, but it can definitely be done.
For example, with 9 coins, divide into 3 groups of 3. Weigh group A against B, if A heavier weigh A against C. You now know which threesome has the fake coin, and whether it's heavier or not, and one more wighing will solve it.
another approach to problem B
